How does the Closure Property prove that the sum of a rational and irrational number is irrational?(1 point)

Question

Responses

Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.

The sum or difference of a rational and irrational number is a rational number.
The sum or difference of a rational and irrational number is a rational number.

The sum or difference of two rational numbers is a rational number.
The sum or difference of two rational numbers is a rational number.

The sum of an irrational and rational number can’t be predicted.

Answer 1

The correct response regarding how the Closure Property proves that the sum of a rational and an irrational number is irrational is:

The sum or difference of a rational and irrational number is a rational number.

However, the statement here is incorrect. More accurately, the sum or difference of a rational and an irrational number is an irrational number. But the provided options do not have the correct wording. The important understanding is that the sum or difference of a rational and an irrational number cannot be a rational number, thus leading to the conclusion that the sum of a rational and an irrational number is indeed irrational.

Answer 2

Mia says that anytime there is a radical in an addition expression, it automatically has an irrational number as the sum. Is she correct?(1 point)
Responses

No, some radicals simplify to rational numbers.
No, some radicals simplify to rational numbers.

Yes, radicals are irrational.
Yes, radicals are irrational.

No, radicals always simplify to integers, so they are rational.
No, radicals always simplify to integers, so they are rational.

Yes, radicals can’t be simplified, so they must be irrational.

Answer 3

The correct response to Mia's statement is:

No, some radicals simplify to rational numbers.

This is true because certain radicals, such as $\sqrt{4}=2$ or $\sqrt{9}=3$, simplify to whole numbers, which are rational. Therefore, having a radical in an expression does not automatically mean that the result of the addition is irrational.

Answer 4

What type of number will result from the expression 19−−√−3/2
?(1 point)
Responses

an irrational number
an irrational number

a rational number
a rational number

an integer
an integer

a whole number

Answer 5

To determine the type of number that results from the expression $\sqrt{19}-\frac{3}{2}$, we can analyze each component:

1. $\sqrt{19}$ is an irrational number because it cannot be simplified to a fraction or a whole number.
2. $\frac{3}{2}$ is a rational number.

Now, when we subtract a rational number ($\frac{3}{2}$) from an irrational number ($\sqrt{19}$), the result remains an irrational number.

Therefore, the correct response is:

an irrational number

Answer 6

Ryiaz claims it is impossible to prove that a rational number multiplied by an irrational number gives an irrational product. How can you prove this?(1 point)
Responses

He’s correct; it is impossible to prove in any way.
He’s correct; it is impossible to prove in any way.

by choosing random numbers to plug into multiplication problems until he’s convinced
by choosing random numbers to plug into multiplication problems until he’s convinced

by using a proof by contradiction to show multiplication is a closed operation for rational numbers
by using a proof by contradiction to show multiplication is a closed operation for rational numbers

by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer

Answer 7

To prove that a rational number multiplied by an irrational number gives an irrational product, you would use a direct proof to demonstrate the property. Therefore, the best response to Ryiaz's claim is:

by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.

In a direct proof, you can assume $r$ is a rational number (which can be expressed as $\frac{a}{b}$, where $a$ and $b$ are integers and $b\ne 0$) and $x$ is an irrational number. The product $r\cdot x=\frac{a}{b}\cdot x$ must also be irrational, because if $r\cdot x$ were rational, then $x$ could be expressed as the ratio of two rational numbers, contradicting the assumption that $x$ is irrational. Hence, this demonstrates that the product of a rational number and an irrational number is indeed irrational.

Answer 8

Find the product of 12−−√
and 5/6
. What type of number is it?(1 point)
Responses

4.1569 . . . ; an irrational number
4.1569 . . . ; an irrational number

2.8867 . . . ; an irrational number
2.8867 . . . ; an irrational number

4.2974 . . . ; an irrational number
4.2974 . . . ; an irrational number

2.6307 . . . ; an irrational number
2.6307 . . . ; an irrational number

Answer 9

To find the product of $\sqrt{12}$ and $\frac{5}{6}$, we first calculate $\sqrt{12}$:

$$\sqrt{12}=\sqrt{4\times 3}=\sqrt{4}\times \sqrt{3}=2\sqrt{3}$$

Now, we multiply this by $\frac{5}{6}$:

$$\text{Product}=2\sqrt{3}\times \frac{5}{6}=\frac{10\sqrt{3}}{6}=\frac{5\sqrt{3}}{3}$$

Since $\sqrt{3}$ is an irrational number, the product $\frac{5\sqrt{3}}{3}$ is also irrational.

Now, let's approximate the numerical value of $\frac{5\sqrt{3}}{3}$:

Using $\sqrt{3}\approx 1.732$:

$$\frac{5\sqrt{3}}{3}\approx \frac{5\times 1.732}{3}=\frac{8.66}{3}\approx 2.8867$$

Therefore, the answer is approximately $2.8867$ and it is an irrational number.

The correct response is:

2.8867 . . . ; an irrational number